Minimum size for the top jet drop from a bursting bubble

Jet drops ejected from bursting bubbles are ubiquitous, transporting aromatics from sparkling beverages, pathogens from contaminated water sources, and sea salts and organic species from the ocean surface to the atmosphere. In all of these processes, the smallest drops are noteworthy because their slow settling velocities allow them to persist longer and travel further than large drops, provided they escape the viscous sublayer. Yet it is unclear what sets the limit to how small these jet drops can become. Here we directly observe microscale jet drop formation and demonstrate that the smallest jet drops are not produced by the smallest jet drop-producing bubbles, as predicted numerically by Duchemin et al. [Duchemin et al., Phys. Fluids 14, 3000 (2002)]. Through a combination of high-speed imaging and numerical simulation, we show that the minimum jet drop size is set by an interplay of viscous and inertial-capillary forces both prior and subsequent to the jet formation. Based on the observation of self-similar jet growth, the jet drop size is decomposed into a shape factor and a jet growth time to rationalize the nonmonotonic relationship of drop size to bubble size. These findings provide constraints on submicron aerosol production from jet drops in the ocean.

Figure 1

Jet drops produced from a bursting bubble. (a) An air bubble of radius $R=1.5$ mm at an air-water interface produces a jet drop of radius $r_d=280\mu \text{m}$ within time $t=4\text{ms}$ after rupturing. (b) Decades of experiments (open symbols), most gathered in Ref. [2], have related the size of the top jet drop $r_d$ to the bubble size $R$ in fresh and salt water at various temperatures [9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]; yet these measurements fall outside the regime in which submicron aerosols from these jet drops might be expected (green shaded box). Results from a computational study [27], included as pluses and dimensionalized to seawater at $20{}^{\circ }\mathrm{C}$, suggest a nonmonotonic dependence of $r_d$ on $R$ in this region. Also plotted are two data points from this study (yellow closed symbols), the smaller of which extends into the submicron regime and has properties similar to salt water at $8{}^{\circ }\mathrm{C}$. The commonly assumed 10% rule (dotted line), an empirical power-law dependence [2] $r_d=150\mu \text{m}{(R/\mathrm{mm})}^{1.3}$ (dashed line), and a theoretical scaling relationship [28] for $20{}^{\circ }\mathrm{C}$ seawater (solid curve) are included, along with regions where no jet drops are expected as a result of either viscous or gravitational effects.

Figure 2

Experimental images of bursting bubbles and resulting jet drops. (a) A microfluidic device provides a precisely designed environment to directly observe jet drops from the rupture of a microbubble. (b) Replacing the surrounding liquid with glycerol-water solutions of varying concentrations primarily changes the viscosity $\mu$ and therefore the Laplace number $\mathrm{La}\equiv \rho \gamma R/{\mu }^2$. Changing the Laplace number within the microfluidic device changes the jet drop size, illustrated here by image subtraction for times before and after rupture. The dashed boxes highlight the first discernible jet drop. (c) Micropipette experiments reveal finer detail of the bubble-bursting process by generating bubbles of radius $R\approx 200\mu \text{m}$ that collapse within hundreds of microseconds. Note that the initial bubble shape in both microfluidic and micropipette experiments is nearly spherical. (d) Changing the glycerol concentration of the liquid in the micropipette experiments reveals a nonmonotonic relationship between the Laplace number and the size of the first discernible jet drop. No jet drops are formed for $\mathrm{La}<320$.

Figure 3

Simulations of jet drops forming from bursting bubbles. (a) Superimposed profiles spanning the formation of the top jet drop demonstrate the nonmonotonic dependence of jet drop size on Laplace number $\mathrm{La}$. Pinch-off times and inversion times are labeled as $t_p$ and $t_0$, respectively, and scaled by the inertial-capillary time $\tau =\sqrt{\rho R^3/\gamma }$. The time between successive interfaces is $0.05\tau$, with the exception of black curves at $t_0$ and $t_p$ and a gray curve at $t_0+0.005\tau$. (b) Close-up view of final stages of cavity collapse, inversion, and jet growth until pinch-off for $\mathrm{La}=1000$, with the time between interfaces scaling as $|t-t_0{|}^{3/2}$ and spanning $0.471\le t/\tau \le 0.486$. The base of the jet at inversion is labeled as $z_0/R$. (c) A self-similar scaling of the same profiles shows an approximate collapse of the growing jet. (d) Close-up of the tip of the jet in self-similar coordinates, with the diameter labeled as $2r^{*}$. (e) The shape factor $r^{*}\equiv r_d{(\rho /\gamma )}^{1/3}{(t_p-t_0)}^{-2/3}$ and jet growth time $t^{*}\equiv (t_p-t_0)/\tau$ extracted from simulations of varying Laplace number. Both relationships are approximated by hockey-stick fits, shown as dashed lines. Symbols with lighter shade denote data points excluded from the fits (see Appendix pp3).

Figure 4

Dimensionless and dimensional plots of jet drop radius relative to bubble radius. (a) Ratio of jet drop radius to bubble radius as a function of Laplace number for both sets of experiments and simulations. The dashed black line corresponds to Eq. (1), the self-similar model prediction based on Fig. 3. The solid green curve denotes the theoretical scaling [28] $r_d/R=k_d{(\sqrt{\mathrm{La}/{\mathrm{La}}_c}-1)}^{5/4}/{\mathrm{La}}^{3/8}$, with $k_d=0.6$ and ${\mathrm{La}}_c=540$. (b) Jet drop radius as a function of bubble radius from previous studies [same symbols as in Fig. 1], along with the redimensionalized model (2) for seawater at temperatures of $0{}^{\circ }\mathrm{C},8{}^{\circ }\mathrm{C}$, and $40{}^{\circ }\mathrm{C}$. The yellow triangle corresponds to the microfluidic data point [Fig. 2] from the 15% glycerol-water solution, as the liquid properties are similar to seawater at $8{}^{\circ }\mathrm{C}$.

Figure 5

Viscous length scale of seawater as a function of temperature. Reference correlations [45, 46] for the viscosity, density, and surface tension of seawater with salinity 35 g ${\mathrm{kg}}^{-1}$ (typical of the oceans) as a function of temperature are combined to form the viscous length scale $R_{\mu }\equiv {\mu }^2/\rho \gamma$. This length scale is plotted along with an Arrhenius fit of the form $R_{\mu }\left(T\right)=Aexp\left(\frac{B}{C+T{/}^{\circ }\mathrm{C}}\right)$, where $A=0.06202$ nm, $B=661.3$, and $C=100.1$. The Arrhenius fit differs from the combined reference correlations by less than 0.5% for the whole temperature range $0{}^{\circ }\mathrm{C}\le T\le 40{}^{\circ }\mathrm{C}$.

Figure 6

Microfluidic device design. (a) Schematic of the microfluidic device with channel flow moving from left to right. Air and liquid enter channels through the three circular ports at the left-hand side of the device. (b) The liquid and air are forced through a $10\text{−}\mu \text{m}$-diameter nozzle where air bubbles are formed. Bubbles produced at the nozzle are of radius $R\approx 30\mu \text{m}$. The channel width and height after the nozzle are $80\mu \text{m}$. (c) Air bubbles come into contact with an air-liquid interface and burst at the device outlet.

Figure 7

Micropipette experimental setup. (a) Schematic of the micropipette experiment. (b) Image of an air bubble in a 20% glycerol-water solution just after pinching off from the micropipette.

Figure 8

Initial bubble shape for varying Bond number $\mathrm{Bo}=\rho g{R}^2/\gamma$. Each bubble has a constant volume-equivalent radius $R$. For simulations including gravitational effects, the bubble is initialized without the spherical film cap (drawn in gray for each bubble).

Figure 9

Escape from pinch-off [35]. (a) For simulations near the minimum in jet drop size, in some cases the jet pinched off relatively early $(\mathrm{La}=1000$, top), while in others the jet initially escaped pinch-off $(\mathrm{La}=1100$, bottom). The neck radius temporarily reverses from shrinking to growing and extra liquid is collected at the tip, resulting in a larger jet drop when it later pinches off. (b) Experiments with bursting bubbles of radius $R=200\mu \text{m}$ in 55% glycerol solutions $(\mathrm{La}=320)$ similarly had some jetting events with an early pinch-off (top) along with others in which the jet escaped pinch-off (bottom), leading to a much larger jet drop. Dashed boxes highlight the pinch-off and escape from pinch-off events in each case.

Figure 10

Overlays of the air-water interface during bubble burst from simulation results (red) with experimental images. The experimental bubble radius is $R=209\mu \text{m}$ and bursts in water, corresponding to a Laplace number $\mathrm{La}=17200$. The time between snapshots is $20\mu \text{s}$ and no fitting parameters are used other than the offset time, here $26\mu \text{s}$, of the first experimental image relative to $t=0$ in the simulation.

Figure 11

Scaling of jet lengths after inversion for $\mathrm{La}<1200$. (a) Snapshot of jet at $t/\tau =0.76$ for $\mathrm{La}=610$ with jet radius $r_j$, jet height $z_j$, base radius $r_b$, base height $z_b$, and base height at inversion $z_0$ labeled accordingly (omitting nondimensionalizing factors of $R)$. The profiles at $t=0$ and the inversion time $t_0$ are plotted in gray and blue, respectively. (b) Scaling of these jet lengths with $(t-t_0)/\tau$ for simulations with $\mathrm{La}=610$, 830, and 1000. Top drop radii are plotted at time of pinch-off with circles and the curve $0.166{[(t-t_0)/\tau ]}^{2/3}$ is the model prediction for self-similar jet drop radius evolution for $\mathrm{La}\lesssim 1200$, as $r^{*}=0.166$ here.

Figure 12

Scaling of jet lengths after inversion for $\mathrm{La}>1200$. (a) In this regime, $0.166{[(t-t_0)/\tau ]}^{2/3}$ is no longer a good estimate for jet tip or drop radius; the model instead assumes a constant scaled jet growth time to pinch-off of $t^{*}=0.04$, shown here as a vertical dotted line. For larger Laplace numbers, the base radius starts at a nonzero value at inversion [as seen in Fig. 3] but follows a ${(t-t_0)}^{2/3}$ scaling at later times. Markers have been added to the first ten time steps after inversion to highlight the temporal resolution of the data; the jump in $r_b$ for $\mathrm{La}=3000$ is due to the inversion of a smaller jet base before the main one. (b) While the jet tip height $z_j$ grows roughly with ${(t-t_0)}^{2/3}$, the base height $z_b$ at large $\mathrm{La}$ appears to grow linearly with time.